TITLE: Simplify $y = \frac{1-2\exp(-x)\cos(x)+\exp(-2x)}{1+2\exp(-x)\sin(x)-\exp(-2x)}$
QUESTION [5 upvotes]: I'm trying to simplify the following equation:
$y = \dfrac{1-2\exp(-x)\cos(x)+\exp(-2x)}{1+2\exp(-x)\sin(x)-\exp(-2x)}$
I suspect that a simpler form using complex exponents exists, but I can't find it.
For context, this equation describes the effective conductivity due to the skin effect of a flat conductor as a function of its thickness. I just removed some scale factors for simplicity. The underlying differential equation gives rise to expressions of the form $\exp(\pm(1+i)x)$, which is where the $\sin(x)$ and $\cos(x)$ came from.

REPLY [9 votes]: After coreyman317's answer and your comment about large values of $x$, you could notice that for $x >24$
$$\frac{\cosh(x)-\cos(x)}{\sinh(x)+\sin(x)} \sim \coth(x)$$ for an error $ < 10^{-10}$.
Moreover, for large $x$, $\coth(x)\sim 1+2e^{-2x}$.